Have you ever heard of a math trail? If you love math and walking, this is the perfect combination of the two. Read on to discover more about math trails and how you can start blazing your own.

What Is a Math Trail?

A math trail is a walk to discover mathematics. And it can be almost anywhere—a neighborhood, a business district or shopping mall, a park, a zoo, a library, or even a government building. The math trail map or guide points to places where walkers formulate, discuss, and solve interesting mathematical problems. 

Anyone can walk a math trail alone, with family, or with another group. Walkers cooperate along the trail as they talk about the math problems. There’s no competition or grading. At the end of the math trail, they have the pleasure of having walked the trail and of having done some interesting mathematics. 

The History of the Math Trail

The earliest math trails appeared in England and in Australia. In 1985, Dudley Blane and his colleagues blazed a trail around the center of Melbourne as a holiday-week activity for families. 

The trail’s mathematical ideas included investigating a circular pattern of bricks in the pavement (to discover the invariance of pi), studying the timetables in a train station, looking at the reflection of a cathedral in a pond (to estimate its height), trying to estimate the speed of water rushing down a spillway, counting the number of windows in a wall of a skyscraper, and looking for patterns in the numbers of post office boxes.

Australian mathematics educators constructed many more trails based on a variety of themes and venues, including preparing for prospecting in a gold rush town, acting as an apprentice keeper in a local zoo, and working on the ship works and sailing boats in a historical nautical village. 

Like any good idea, the idea of a math trail has spread and people have adopted it. Carole Greenes of Boston University created a historical mathematics trail in Boston centered on the Common and the Public Garden. Unlike Blane’s Melbourne trail, walkers on Greenes’ trail followed a human guide who knew the historical and mathematical aspects of the trail and who could give hints and suggestions to walkers who got stuck on a task or idea. 

Kay Toliver, an award-winning New York City schoolteacher, leads her students on walks while guiding them to discover mathematics in their school neighborhood. Student walkers do not write their ideas and solutions on paper, but informally discuss their discoveries on the spot and then take the discussion back to the classroom.

Characteristics In the Math Trail Model

While there is a basic model for a math trail, you can adapt it to your location and match your interests. But to get you started, here are some common characteristics of the math trail model:

• Math trails are for everyone. Everyone studies (or studied) math in school. Everyone uses math. Math trail problems should be interesting and accessible to people at all levels of age and experience. 
• Math trails are cooperative, not competitive. In the spirit of the NCTM Curriculum Standards, the emphasis is on talking about and doing mathematics. The purpose of the math trail problems is to bring attention to the processes for formulating and solving problems, not to find single correct solutions. 
• Math trails are self-directed. Such trails are ready when a walker is ready. There is no time limit.
• Math trails are voluntary. An important characteristic of our model is that walking a math trail is entirely voluntary. If a particular bit of mathematics or the setting is not attractive or interesting, it won’t work for a math trail.
• Math trails are opportunistic. We believe that “math is everywhere.” Trailblazers can prove the assertion by taking advantage of their locale: neighborhood street, business district, parking lot, college campus, shopping mall, park, zoo, library, grocery store, clothing store, and more. Any public place that allows safe walking is ripe with math problems for an imaginative trailblazer.
• Math trails are temporary. Places change. Permanent trails require maintenance and continuing time and energy and, perhaps, miss new opportunities. Rather than maintain this year’s trail, blaze a new trail next summer as a way to bring back satisfied walkers and use the novelty to attract new walkers.

How to Start Blazing a Math Trail

Are you ready to get started on your first math trail? These resources will help:

1. Download the “Math Trails” eBook by Joel Schneider, Henry Pollak, and Mary Margaret Shoaf (free for COMAP members). You can also order a print copy here.
2. Watch the 10-minute video about math trails on The Future Channel.
3. Read Hitting the Math Trail on Education World for more math trail tips. 

Have a Great Math Trail Story?

Submit your math trail story here to be considered for a future article on this blog. Of course, we at COMAP are particularly interested in trails that feature some mathematical modeling or optimization problems, but any math students find is good math. Happy trailblazing!